A220417 -id:A220417 - cp's OEIS frontend

A007925 a(n) = n^(n+1) - (n+1)^n.

-1, -1, -1, 17, 399, 7849, 162287, 3667649, 91171007, 2486784401, 74062575399, 2395420006033, 83695120256591, 3143661612445145, 126375169532421599, 5415486851106043649, 246486713303685957375, 11877172892329028459041, 604107995057426434824791

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

From Mathew Englander, Jul 07 2020: (Start)
All a(n) are odd and for n even, a(n) == 3 (mod 4); for n odd and n != 1, a(n) == 1 (mod 4).
The correspondence between n and a(n) when considered mod 6 is as follows: for n == 0, 1, 2, or 3, a(n) == 5; for n == 4, a(n) == 3; for n == 5, a(n) == 1.
For all n, a(n)+1 is a multiple of n^2.
For n odd  and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs of the above, see the Englander link. (End)

			a(2) = 1^2 - 2^1 = -1,
a(4) = 3^4 - 4^3 = 17.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

T. D. Noe, Table of n, a(n) for n = 0..100
Mathew Englander, Notes on OEIS A007925
Sergio Silva, Teste Numerico, Item 3.
H. J. Smith, Contour Plot of z = x^y - y^x

Cf. A051442.

Cf. A166326, A099498, A141074, A174379, A123206, A045575, A082754.

A007925:=n->n^(n+1)-(n+1)^n: seq(A007925(n), n=0..25); # Wesley Ivan Hurt, Jan 10 2017

lst={};Do[AppendTo[lst, (n^(n+1)-((n+1)^n))], {n, 0, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
#^(#+1)-(#+1)^#&/@Range[0,20] (* Harvey P. Dale, Oct 22 2011 *)

A007925[n]:=n^(n+1)-(n+1)^n$ makelist(A007925[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n)=n^(n+1)-(n+1)^n \\ Charles R Greathouse IV, Feb 06 2017

Asymptotic expression for a(n) is a(n) ~ n^n * (n - e). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
From Mathew Englander, Jul 07 2020: (Start)
a(n) = A111454(n+4) - 1.
a(n) = A055651(n, n+1).
a(n) = A220417(n+1, n) for n >= 1.
a(n) = A007778(n) - A000169(n+1).
(End)
E.g.f.: LambertW(-x)/((1+LambertW(-x))*x)-LambertW(-x)/(1+LambertW(-x))^3. - Alois P. Heinz, Jul 04 2022

A051128 Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 11, 100, 729, 4096, 16807, 46656, 78125, 65536, 19683, 1024, 1

Offset: 1

N. J. A. Sloane

Sum of antidiagonals is A003101(n) for n>0. - Alford Arnold, Jan 14 2007

			Table begins
1,    1,    1,    1,    1, ...
2,    4,    8,   16,   32, ...
3,    9,   27,   81,  243, ...
4,   16,   64,  256, 1024, ...

T. D. Noe, Rows n=1..50 of triangle, flattened
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.

Cf. A051129, A003992, A004248, A033918, A095891, A220415, A220416, A220417, A003101.

A051128 := proc(n)       # Boris Putievskiy's formula
a := floor((sqrt(8*n-7)+1)/2);
b := (a+a^2)/2-n;
c := (a-a^2)/2+n;
(b+1)^c end:
seq(A051128(n), n=1..61);  # Peter Luschny, Dec 14 2012
# second Maple program:
T:= (n, k)-> n^k:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Table[n^(k - n + 1), {k, 1, 11}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Dec 14 2012 *)

T(n,k) = n^k \\ Charles R Greathouse IV, Feb 09 2017

a(n) = A004736(n)^A002260(n) or ((t*t+3*t+4)/2-n)^(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012

More terms from James Sellers, Dec 11 1999

A231883 Number of ways to write n = x + y (x, y > 0) with x^2 + (n-2)*y^2 prime.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 4, 1, 5, 2, 5, 1, 4, 4, 3, 1, 7, 2, 3, 3, 6, 7, 3, 2, 6, 2, 9, 3, 8, 3, 10, 3, 5, 8, 8, 4, 7, 5, 13, 4, 12, 6, 7, 6, 8, 10, 14, 4, 17, 9, 9, 6, 9, 5, 8, 5, 9, 7, 12, 10, 11, 7, 11, 8, 12, 4, 13, 3, 22, 6, 16, 7, 14, 8, 10, 4, 14, 4, 17, 9, 16, 6, 12, 11, 14, 4, 21, 4, 21, 8, 18, 3, 11, 14, 23, 7, 22, 5, 23, 8

Offset: 1

Zhi-Wei Sun, Nov 21 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 8, 12, 16. Moreover, if m and n are positive integers with m >= max{2, n-1} and gcd(m, n+1) = 1, then x^2 + n*y^2 is prime for some positive integers x and y with x + y = m, except for the case m = n + 3 = 29.
(ii) Let m and n be integers greater than one with m >= (n-1)/2 and gcd(m, n-1) = 1. Then x + n*y is prime for some positive integers x and y with x + y = m.
(iii) Any integer n > 3 not equal to 12 or 16 can be written as x + y (x, y > 0) with (n-2)*x - y and (n-2)*x^2 + y^2 both prime.

			 a(8) = 1 since 8 = 5 + 3 with 5^2 + (8-2)*3^2 = 79 prime.
a(12) = 1 since 12 = 11 + 1 with 11^2 + (12-2)*1^2 = 131 prime.
a(16) = 1 since 16 = 15 + 1 with 15^2 + (16-2)*1^2 = 239 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A036468, A219842, A219864, A220417, A232174, A232186, A232194.

a[n_]:=Sum[If[PrimeQ[x^2+(n-2)*(n-x)^2],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A007925 a(n) = n^(n+1) - (n+1)^n.

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A051128 Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).

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Maple

Mathematica

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Extensions

A231883 Number of ways to write n = x + y (x, y > 0) with x^2 + (n-2)*y^2 prime.

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Keywords

Comments

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Mathematica